Download Algebraic numbers of small Weil`s height in CM

Journal of Number Theory 122 (2007) 247–260
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Algebraic numbers of small Weil’s height in CM-fields:
On a theorem of Schinzel
Francesco Amoroso a,∗ , Filippo A.E. Nuccio b
a Laboratoire de Mathématiques “N. Oresme”, UMR 6139 (CNRS), Université de Caen,
Campus II, BP 5186, F-14032 Caen cedex, France
b Dipartimento di Matematica “G. Castelnuovo”, Università “La Sapienza”,
Piazzale Aldo Moro, 2 I-00185 Roma, Italy
Received 1 August 2004; revised 3 February 2006
Available online 26 May 2006
Communicated by David Goss
Abstract
Let S be the union of all CM-fields and S 0 be the set of non-zero algebraic numbers of S which are not
roots of unity. We show that in S 0 Weil’s height cannot be bounded from below by an absolute constant.
© 2006 Elsevier Inc. All rights reserved.
1. Introduction
Let K be a CM-field. A. Schinzel proved [Sch 1973] that the Weil height of non-zero algebraic
numbers in K is bounded from below by an absolute constant C outside the set of algebraic
numbers such that |α| = 1 (since K is CM, |α| = 1 for some archimedean place guarantees
|α| = 1 for all archimedean places). More precisely, his result reads as follows: for any α ∈ K× ,
|α| = 1, we have
√ 1+ 5
1
h(α) C := log
.
2
2
* Corresponding author.
E-mail addresses: [email protected] (F. Amoroso), [email protected] (F.A.E. Nuccio).
0022-314X/$ – see front matter © 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.jnt.2006.04.005
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F. Amoroso, F.A.E. Nuccio / Journal of Number Theory 122 (2007) 247–260
E. Bombieri and U. Zannier, motivated also by the above result, asked (private communication to
the first author) for an absolute lower bound for the height of non-zero algebraic numbers lying
in a complex abelian extension outside the set of roots of unity. This question was solved by
R. Dvornicich and the first author (see [AmoDvo 2000]), who proved that for any α in a complex
abelian extension, α = 0 and α not a root of unity, we have
h(α) Cab :=
log 5
.
12
In the same paper it was shown that Cab cannot be replaced by any constant > (log 7)/12,
since there exists an element α in a cyclotomic field such that h(α) = (log 7)/12; we note that
(log 7)/12 < C.
One may now ask whether the abelianity condition is necessary for this latter lower bound
or if this bound holds in general for CM-fields. As a first step towards an answer, in his Master
thesis [Nuc 2004] the second author uses the classification of all dihedral principal CM-fields
given by S. Louboutin and R. Okazaky (see [LouOka 1994]) to prove the following:
Theorem 1.1. There exists a normal CM-field L (of degree 8 and such that Gal(L/Q) ∼
= D4 , the
dihedral group of order 8) whose ring of integers OL contains an element γ of height
h(γ /γ̄ ) =
log |NQL (γ )|
[L : Q]
=
log 2
< Cab .
8
Motivated by this example, we formulate the following natural question in this context:
Question 1.2. Does there exist an absolute constant CCM ∈ (0, Cab ) such that for every CM-field
K and for every α ∈ K× \ Ktors , h(α) > CCM holds?
We prove the following theorem, which gives a negative answer to the previous question:
Theorem 1.3. There exists an infinite sequence (αk ) of algebraic numbers such that the fields
Q(αk ) are CM-fields, αk is not a root of unity, dk = [Q(αk ) : Q] → +∞ and
h(αk ) ∼
log(dk )
= o(1)
dk
as k → +∞.
As an application of the main theorem in [AmoDvo 2000], a lower bound for the norm of
algebraic integers γ such that γ /γ̄ is not a root of unity was given (see [AmoDvo 2000, Corollary 1]):
Theorem (Amoroso–Dvornicich). Let γ be an integer lying in an abelian extension L of Q.
Then, if γ /γ̄ is not a root of unity,
log |NQL (γ )|
[L : Q]
log 5
.
12
F. Amoroso, F.A.E. Nuccio / Journal of Number Theory 122 (2007) 247–260
249
Our result allows us to show that also this bound is not anymore true if the abelianity condition
is dropped (see Theorem 5.1).
Our proof relies on elementary facts about reciprocal polynomials and on an operator δ introduced by the first author (see [Amo 1995]). We will construct two families of polynomials
having all their roots on the unit circle and with “small” leading coefficient. We will eventually
show that these polynomials are either irreducible or have a (non-monic) irreducible factor of
high degree: this will ensure their roots have small height. Moreover, having all their roots on the
unit circle, the fields defined by those polynomials are CM-fields.
The paper is organized as follows: in Section 2 we recall some basic facts on Weil’s height
and on CM-fields and we introduce δ. Section 3 is devoted to the dihedral example. In the two
following sections, we produce the two families mentioned above. Finally, in the last section, we
propose a conjecture about polynomials defining CM-fields.
2. Auxiliary results
2.1. Weil’s height
Let α ∈ Q and let K be a number field containing α. We denote by MK the set of places of K.
For v ∈ K, let Kv be the completion of K at v and let | · |v be the (normalized) absolute value of
the place v. Hence, if v is an archimedean place associated with the embedding σ : K → Q
|α|v = |σ α|,
and, if v is a non-archimedean place associated to the prime ideal p over the rational prime p,
|α|v = p −λ/e ,
where e is the ramification index of p over p and λ is the exponent of p in the factorization of
the ideal (α) in the ring of integers of K. This standard normalization agrees with the product
formula
v :Qv ] = 1
|α|[K
v
v∈MK
which holds for α ∈ K× . We define the (Weil) height of α by
h(α) =
1
[Kv : Qv ] log max |α|v , 1 .
[K : Q]
v∈MK
It is easy to check that h(ζ ) = 0 for every root of unity ζ ∈ Q: it is in fact equivalent to Kronecker’s theorem.
For later use, we also need (see, for instance, [Wal 2000, Chapitre 3]):
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F. Amoroso, F.A.E. Nuccio / Journal of Number Theory 122 (2007) 247–260
Remark 2.1. Let α be a non-zero algebraic numbers and let P (X) ∈ Z[X] be its minimal polynomial over the integers, i.e. P (α) = 0 and P is irreducible of leading coefficient > 0. Let
K = Q(α); then
log =
[Kv : Qv ] log max |α|v , 1 .
v∈MK ,v∞
2.2. Reciprocal polynomials and CM-fields
We now recall some basic facts about reciprocal and antireciprocal polynomials.
Definition 2.2. Let P (X) ∈ C[X] be a polynomial of degree d. If P (X) = X d P (X −1 ), then P is
said to be reciprocal. If P (X) = −X d P (X −1 ), it is said to be antireciprocal.
Let us emphasize that the factors of a reciprocal or antireciprocal polynomial with real coefficients should have specific form. Firstly, if P (X) ∈ R[X] is a reciprocal (or antireciprocal)
polynomial of degree d and α is a root of P , then α = 0 and α −1 is still a root of P . If now
P (±1) = 0, the distinct values {α, α −1 } are two roots of the polynomial having the same multiplicity and the polynomial is reciprocal of even degree, all of its roots being “coupled.” We can
then factorize a general reciprocal (or antireciprocal) polynomial P as
P (X) = (X − 1)a (X + 1)b Q(X),
where Q(X) is reciprocal not vanishing at ±1 of degree 2k and d = a + b + 2k. Moreover,
a ≡ 1 mod 2 if P is antireciprocal and b ≡ 1 mod 2 if P is reciprocal of odd degree or if it is
antireciprocal of even degree.
A totally imaginary quadratic extension K of a totally real number field K+ is said to be
a CM-field. As mentioned in the introduction, one of the main properties of CM-fields is the
following: let α ∈ K and assume that |τ α| = 1 for some embedding τ ; then for any embedding
σ we have |σ α| = 1. Indeed, the complex conjugation induces an automorphism on K which
is independent of the embedding into C (see [Was 1982, p. 38]). The following characterizes
CM-fields and will be useful in our main construction:
Proposition 2.3. A number field K = Q is CM if and only if there exists a monogenic1 element
α ∈ K such that |σ α| = 1 for all embedding σ .
Proof. Let K be a CM-field and let γ ∈ K be a monogenic element. For n ∈ Z we put
αn =
γ +n
.
γ̄ + n
1 We say that α in a number field K is monogenic with respect to K when K = Q(α).
F. Amoroso, F.A.E. Nuccio / Journal of Number Theory 122 (2007) 247–260
251
Clearly, |σ αn | = 1 for every σ . In order to show that some αn is monogenic, let us point out
that there are two different integers n, m ∈ Z such that αn = αm and Q(αn ) = Q(αm ), since the
number of subfields of K is finite. Therefore,
γ=
mαn (1 − αm ) − nαm (1 − αn )
∈ Q(αn )
αm − αn
and K = Q(αn ).
Conversely, let us assume that K = Q(α) where |σ α| = 1 for all embedding σ . Put
K+ = Q α + α −1 ,
then K+ is a totally real field and [K : K+ ] = 2, because α ∈
/ R and K = Q.
2
Thus, a CM-field can be defined by an irreducible polynomial P ∈ Q[X] vanishing only on the
unit circle. We also remark that such a polynomial must be reciprocal, since |α| = 1 ⇒ ᾱ = α −1 .
2.3. A differential operator
We now introduce an operator δ (see [Amo 1995]) over C[X] which has all the properties of
a derivation but linearity: if P (X) ∈ C[X] is a polynomial with complex coefficients of degree d
(we set deg(0) = 0), we define
δ(P )(X) = X
d
dP
(X) − P (X).
dX
2
It is obvious that, if we denote by pd the leading coefficient of P , the leading coefficient of
δ(P ) is dpd /2, and that δ(P ) and P have the same degree. Moreover, a classical property of a
derivation is satisfied:
δ(P Q) = δ(P )Q + δ(Q)P ;
therefore, for n ∈ N,
δ P n = nP n−1 δ(P ).
The following remark (due to D. Simon) can also be useful. Let D the derivation
D(F ) =
1
∂F
∂F
X
−Y
2
∂X
∂Y
on C[X, Y ]. Then for any P ∈ C[X] we have δ(P )(X) = D(h P )(X, 1), where h P (X, Y ) =
Y deg(P ) P (X/Y ) is the homogenization of P .
Having recalled all these basic facts, we can prove the main property of the operator δ (see
also [Amo 1995, Proposition 1]):
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F. Amoroso, F.A.E. Nuccio / Journal of Number Theory 122 (2007) 247–260
Lemma 2.4. Let P (X) ∈ C[X] be a reciprocal polynomial (respectively antireciprocal) having all its roots on the unit circle. Then δ(P )(X) is an antireciprocal (respectively reciprocal)
polynomial whose roots still lie on the unit circle. Moreover, if α1 , . . . , αk are the distinct roots
of P (X) of multiplicity m1 , . . . , mk , then δ(P )(X) vanishes at αj with multiplicity mj − 1 for
j = 1, . . . , k, and at certain β1 , . . . , βk with multiplicity equal to 1. Finally, the set {α1 , . . . , αk }
and {β1 , . . . , βk } are intercalated.2
Proof. We begin by showing that δ transforms reciprocal polynomials into antireciprocal ones,
and vice versa: directly from the definition, it is clear that the condition of being reciprocal
(respectively antireciprocal) for a polynomial
P (X) =
d
aj X j
j =0
turns out to be aj = ad−j (respectively aj = −ad−j ). Looking closely at the definition of δ and
setting
δ(P )(X) =
d
bj X j ,
j =0
it is clear that for j = 1, . . . , d we have bj = (j − d/2)aj and therefore
bd−j = (d − j − d/2)ad−j = (d/2 − j )ad−j .
Thus, when P is reciprocal, aj = ad−j implies bj = −bd−j and so δ(P ) is antireciprocal; and
when P is antireciprocal, aj = −ad−j implies bj = bd−j and so δ(P ) is reciprocal.
Let us now suppose P to be a reciprocal polynomial of degree d and let αj = eiϑj : then
d
f (t) = e−i 2 t P eit
d
is a periodic real function (in fact f (t) = ei 2 t P (e−it ) = f (t)) of period equal to 2π which vanishes at {ϑ1 , . . . , ϑk } ⊂ [0, 2π). Thanks to Rolle’s theorem, f (t) vanishes at certain {φ1 , . . . , φk }
and the sets
eiϑ1 , . . . , eiϑk
and
eiφ1 , . . . , eiφk
are intercalated. But
d −i d t it e 2 P e
dt
it
d dP (e )
d
d
− i e−i 2 t P eit
= e−i 2 t
dt
2
f (t) =
2 Two finite sets S, T ⊂ {z ∈ C, |z| = 1} are intercalated if they have the same cardinality and if there is one, and only
one, α ∈ S between each pair of consecutive β, β ∈ T along the unit circle.
F. Amoroso, F.A.E. Nuccio / Journal of Number Theory 122 (2007) 247–260
253
d
d
d
= ieit−i 2 t P eit − i e−i 2 t P eit
2
d it −i d2 t
it it
e P e − P e
= ie
2
d
= ie−i 2 t δ(P ) eit ,
(2.1)
and so {eiφ1 , . . . , eiφk } are roots of δ(P ). But, if eiϑj is a root of P (X) having multiplicity mj ,
then eiϑj is a root of P (X) having exact multiplicity equal to mj − 1; hence we see, from (2.1),
that eiϑj is a root of δ(P )(X) having multiplicity mj − 1. Finally deg(δ(P )) = deg(P ) = d, and
so the relation
d = k + (m1 − 1) + · · · + (mk − 1)
shows that δ(P )(X) can have no other roots, and that {eiφ1 , . . . , eiφk } are simple roots.
If P were antireciprocal the same argument would lead to our conclusion, using the function
g(t) = if (t). 2
3. A dihedral example
The main idea for this construction is the relation between the norm and the height of certain
algebraic numbers pointed out in [AmoDvo 2000, Corollary 1]: if γ is an algebraic integer of a
CM-field K such that the ideals (γ ) and (γ̄ ) are coprime (and then γ /γ̄ is not a root of unity),
thus
h(γ /γ̄ ) =
log |NQK (γ )|
[K : Q]
(3.1)
.
Working in principal fields one has elements of norm p for each prime p | p, provided its inertial
degree equals 1, and relation (3.1) may therefore be easily employed. We shall then consider
(letting notations be as in [LouOka 1994]), the fields Lp,q , which are the unique cyclic quartic
√
extensions of Q( pq ) unramified at the finite places and are dihedral octic CM-fields. In the
sequel eFK (f), fFK (f) (or simply e(f), f (f) when there is no ambiguity) will denote ramification index and inertial degree, respectively, of a prime ideal f ⊆ OK in an extension K/F: moreover, we
will feel free to write e(), f () for some rational prime when considering normal extensions
K/Q.
Looking for elements whose height is smaller than the constant Cab quoted in the introduction,
we want
log log 5
<
,
8
12
√
Q( pq )
that forces = 2; since we should require f (2) = 1 in order to use (3.1), we need fQ
(2)=1,
i.e. either p, q ≡ 1 mod 2 or p = 2 and q ≡ 1 mod 2. Comparing these conditions with the list
of all dihedral octic principal CM-fields given in paper [LouOka 1994], the only fields we can
consider are the Lp,q with p < q and
(p, q) ∈ (2, 17), (2, 73), (2, 89), (2, 233), (2, 281), (17, 137), (73, 97) .
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F. Amoroso, F.A.E. Nuccio / Journal of Number Theory 122 (2007) 247–260
L
Claim 3.1. fQ 2,17 (2) = 1.
We shall prove Claim 3.1 later on, and use it now to prove Theorem 1.1: from now on, we set
L := L2,17 .
√
L (2) = 2, because L/Q( 34 ) is unramified at the finite
Proof of Theorem 1.1. First of all, eQ
places; indeed, Claim 3.1 implies that the prime 2 is factorized in OL as
2OL = (p1 p2 p3 p4 )2
and we will fix the four primes {pi }4i=1 from now on. Let us emphasize that having Gal(L/Q) ∼
=
D4 , the Galois structure of the extension is as follows:
L
2
K1
K2
√ √
Q( 2, 17 ) := L+
F1
F2
2
√
Q( 2 )
√
Q( 34 )
√
Q( 17 )
2
Q
where K1 ∼
= K2 F1 ∼
= F2 are non-normal quartic CM-fields. If Di denotes the decomposition
subgroup at pi for i = 1, . . . , 4, the conditions f (2) = 1 and e(2) = 2 imply that |Di | = 2 for
i = 1, . . . , 4. We need only to show that for at least one (and then for all) index i, the complex
conjugation χ does not belong to Di , because then the ideals pi and χ(pi ) = pi will be coprime.
that pi ∩ OLi
If Li := LDi are the decomposition subfields and we fix an index i, it is well known
√
is unramified in Li /Q and that e(pi ∩ OLi ) = e(pi ) = 2 in L/Li : since L/Q( 34 ) is unramified
at the finite places, we cannot have
√
L ⊃ Li ⊃ Q( 34 )
F. Amoroso, F.A.E. Nuccio / Journal of Number Theory 122 (2007) 247–260
255
and therefore Li = L+ = L{e,χ} , showing that χ ∈
/ Di for all i. Therefore, if (γ ) = p1 , the ideals
(γ ) and (γ ) are coprime, and (3.1) gives
h(γ /γ̄ ) =
log |NQL (γ )|
[L : Q]
=
log 2
< Cab .
8
2
Concerning Claim 3.1, an easy computation made with Pari-GP allows one to obtain much
L
more: (2, 17) is the only pair of primes such that fQ p,q (2) = 1. But within the scope of our
construction, it is enough to show that the inertial degree of (2) in L2,17 equals 1 (since examples
in other fields would not have smaller height, because [Lp,q : Q] = 8 for all (p, q)) and this can
be shown in a much more theoretical way, as follows.
Theorem 7 of [LouOka 1994] explicitly gives the construction of Lp,q , defining them as the
√
unique cyclic quartic extension of Q( qp ) unramified at finite places. In particular, they are
√
precisely the ray class fields for the modulus m = ∞ of Q( qp ): therefore, Class Field Theory
√
gives an isomorphism between their Galois groups over Q( qp ) and the narrow class group C+
√
of Q( qp ) via Artin reciprocity map:
∼
√
=
ϕ : C+ −→ Gal Lp,q /Q( qp ) ,
√
p −→ p, Gal Lp,q /Q( qp ) ,
√
where (p, Gal(Lp,q /Q( qp ))) is the Frobenius of the prime p in the abelian extension
√
Lp,q /Q( qp ). Since inertial degree of a prime coincides with the order of its Frobenius,
√
it suffices to verify that the primes over (2) in Q( pq ) become trivial in the narrow class
group—i.e. they have a totally positive generator—to establish Claim 3.1. Throughout we fix
[Coh √
1980]) that
(p, q) = (2, 17) and
√ we set L :=√L2,17 . It is well
√ known (see, for instance,
√
(2)OQ(√34 ) = (2, 34 )2 : but (2, 34 ) = (6 + 34 ), because 2 = (6 + 34 )(6 − 34 ) while
√
√
√
√
√
34 = (6 + 34 )(−17 + 3 34 ), and (6 + 34 ) is totally positive, so that ϕ(6 + 34 ) = 1
and Claim 3.1 is established. Actually, the same computation may be performed to show that L
is the only field satisfying our condition, but it becomes much longer: the key point is that 34 is
the only product pq of primes in our list being ≡ −2 modulo a perfect square.
4. A first family
We are now ready to produce the first family of polynomials that we have mentioned in the
introduction.
Proof of Theorem 1.3. For n ∈ N, n 2 we put
Φn (X) =
φm (X)
mn
(where φm (X) is the mth cyclotomic polynomial) and let 2s(n) = 2s denote its degree. An elementary estimate (see, for instance, [HarWri 1938, Theorem 330]) gives
2s =
3n2
+ O(n log n).
π2
(4.1)
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F. Amoroso, F.A.E. Nuccio / Journal of Number Theory 122 (2007) 247–260
According to Bertrand’s Postulate (op. cit., Theorem 418), for all n there exists a natural integer
r < s such that r + s := is a prime number: we define
Rn (X) = (X − 1)2r Φn (X) ∈ Z[X].
Claim 4.1. At most one cyclotomic polynomial φν (X) = X − 1 may divide δ(Rn ), in which case
n ν c1 n log n, where c1 is some positive absolute constant.
Lemma 2.4 gives φν Rn for ν = 2, . . . , n. On the other hand, assume ν > c1 n log n and let p
be the smallest prime not dividing ν. Then, by elementary number theory, p c2 log n for some
absolute constant c2 > 0. Therefore, if c1 is sufficiently large, ν/p > n. If we had φν | δ(Rn ), then
the polynomial δ(Rn ) would have two roots e2πi/ν and e2pπi/ν lying on the unit circle and such
that no root of Rn would lie between them all along the unit circle (since 0 < 1/ν < p/ν < 1/n),
thus contradicting Lemma 2.4. By the same argument, if we had φl φν | δ(Rn ), with l > ν, we
would find two roots {e2πi/ν , e2πi/ l } having no root of Rn between them, which is also absurd
by Lemma 2.4. Finally, again by the same lemma, φν2 δ(Rn ) for ν > n, because of the bound for
roots multiplicity of δ(Rn ). Claim 4.1 is then established.
We may therefore factorize δ(Rn ) as
δ(Rn )(X) = (X − 1)2r φν (X) Pn (X),
where n < ν < c1 n log n and ∈ {0, 1}. Moreover, the leading coefficient δ(Rn ) is equal to the
prime = r + s, and so Pn (X) is an irreducible polynomial of degree d, where
d = 2s − ϕ(ν) ∈ (2s − c1 n log n, 2s].
√
But s < 2s and thus, thanks to (4.1), = d + O( d ): then we have
log ∼ log d
for n → +∞. Let αn be a root of Pn ; the field
Kn = Q(αn )
is a CM-field (by Proposition 2.3). Moreover, since the only contribution to Weil’s height of αn
(of absolute value 1) comes from the non-archimedean places and since Pn has leading coefficient ,
h(αn ) =
(see Remark 2.1).
log log d
∼
d
d
2
5. A second family
Although the preceding construction already shows the impossibility of finding an absolute
lower bound for Weil’s height in non-necessarily normal CM-fields (the case of a normal nonabelian CM-field being still open), we shall present a second family of polynomials which shows
F. Amoroso, F.A.E. Nuccio / Journal of Number Theory 122 (2007) 247–260
257
that also the bound given by the theorem of Dvornicich and the first author quoted in the introduction is not anymore true if the abelianity condition is dropped.
Theorem 5.1. There exists a sequence (αp ) (p prime 5) of algebraic numbers such that:
• the fields Ep = Q(αp ) are CM-fields of degree p − 1;
• we have h(αp ) = (log p)/(p − 1);
• γp = 1/(αp − 1) is an algebraic integer, γ̄p /γp = −αp and
E
NQp (γp ) = p.
Proof. Let p 5 be a prime number. Since (X − 1)φ2p (X) is antireciprocal of odd degree,
Lemma 2.4 and the remarks following Definition 2.2 ensure that
Rp (X) :=
2δ((X − 1)φ2p (X))
X+1
is a polynomial with integer coefficients, not vanishing at ±1.
Claim 5.2. The polynomials Rp are irreducible.
Reducing Rp mod p is an easy task, which we can fulfill by simply pointing out that
φ2p (X) ≡ (X + 1)p−1
mod p
and so
(p − 1)(X + 1)p−2 2δ(X + 1) ≡ 2δ φ2p (X) ≡ −(X + 1)p−2 (X − 1)
mod p,
which finally gives
(X + 1)Rp (X) = 2δ(X − 1)φ2p (X) + 2(X − 1)δ φ2p (X)
≡ (X + 1)p − (X + 1)p−2 (X − 1)2
≡ 4X(X + 1)
p−2
mod p
mod p
and the factorization of Rp [X] ∈ Fp [X] is
Rp (X) ≡ 4X(X + 1)p−3
mod p.
(5.1)
Since the leading coefficient of Rp is precisely p, the irreducibility of Rp will follow as soon
as we show that it has no cyclotomic factors. Indeed, let us suppose that φn (X) | Rp (X). Since
Rp (±1) = 0, we have n 3; but then Eq. (5.1) forces the condition φn (−1) ≡ 0 mod p. It
is well known (see [Apo 1970]) that this condition is verified if and only if n = 2p m , while
ϕ(n) = deg(φn ) deg(Rp ) = p − 1 implies n = 2p, which is absurd by Lemma 2.4. Claim 5.2
follows.
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F. Amoroso, F.A.E. Nuccio / Journal of Number Theory 122 (2007) 247–260
The arguments of Section 4 now imply that the polynomials Rp define CM-fields Ep of degree p − 1 which contain elements αp (the roots of Rp (X)) whose heights are
log(p)
.
p−1
h(αp ) =
We now prove that γp = 1/(αp − 1) is an algebraic integer of norm p. Indeed, γp is a root of
Fp (X) = X
p−1
Rp
1
1+
.
X
The leading coefficient of Fp is
Fp (X)
1
1
+
= Rp (1);
=
lim
R
p
X→+∞ X p−1
X→+∞
X
lim
but we have
Rp (X) = φ2p (X) + 2
X−1 δ φ2p (X)
X+1
⇒
Rp (1) = φ2p (1) = 1,
and so γp ∈ OEp . Concerning its norm, we begin writing
Rp (X) = p
p−1
(X − αi ),
i=1
then
E
NQp (γp ) =
p−1
(αi − 1)−1 = pRp (1)−1 = p.
i=1
We finally remark that γ̄p /γp = (αp − 1)/(ᾱp − 1) = −αp , since |αp | = 1.
2
Remark 5.3. The existence of the element γp shows that in the trivial class of the ideals class
group of Ep there are always two primes over p. More precisely, it could be proved (using, for
instance, the results of [DelDvoSim 2004]) that the prime p splits in the ring of integers of Ep as
p−3
p1 p2 p3
,
where p1 = (1/(αp − 1)) and p2 = p1 = (αp /(αp − 1)): the fields Ep are therefore far from being
normal over Q.
F. Amoroso, F.A.E. Nuccio / Journal of Number Theory 122 (2007) 247–260
259
6. A conjecture on polynomials defining CM-fields
Let
Cycl = φ1e1 · · · φkek , such that k ∈ N and e1 , . . . , ek ∈ N
be the set of products of cyclotomic polynomials. The possibility of finding these families of
CM-fields defined by polynomials in the image
δ(Cycl)
leads us to suppose that every CM-field may be obtained in such a fashion. Some computations
lead us to propose the following conjecture:
Conjecture 6.1. Let K be a CM-field defined by a root of an irreducible polynomial P ∈ Z[X].
∈ Cycl and a rational r such that
Then there exist Φ, Φ
.
δ(Φ) = r ΦP
Moreover, we can perhaps choose r = ±1.
∈ Cycl and a rational r such that
Let P as in Conjecture 6.1 and assume that there exist Φ, Φ
.
δ(Φ) = r ΦP
m (X) = Φ(X
m ). Then the polyLet m ∈ N and put Pm (X) = P (X m ), Φm (X) = Φ(X m ) and Φ
nomial Pm defines again a CM-field and
m Pm ,
δ(Φm ) = rmΦ
which gives some evidence to Conjecture 6.1.
Acknowledgments
We are indebted to D. Simon for a useful discussion about Theorem 5.1. The second author is
also grateful to J. Cougnard for enlightening comments on Section 3. We are finally indebted to
the Referee for several useful suggestions, especially for the remark following Conjecture 6.1.
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